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Differential Equations for Applications
Math 3410, Sections 001 and 002  Elementary Differential Equations, Fall 2017 [
Course Syllabus]
Lectures: MWF 09:05  09:55 at LH 201.
Office hours: Monday 12:00  13:30, Wednesday 12:30  14:00 at MONT 304.
Lecture Notes
 Note 1  Note 2  Note 3  Note 4  Note 5(We skipped the applications and we will come back to that later).
 Note 6 is about Autonomous Equations and Stability of Equilibrium Solutions.
 Note 7 is about Secon Order Linear equations and Laplace Transform.
 Note 8 is about Power series solutions of DEs.
 Note 9 is about Ordinary and Regular points and Method of Frobenious.
 Note 10 is about Bessel Equation.
 Note 11 is about Boundary Value Problems and Fourier Series.
 Note 12 is about Partial Differential Equations.
 Note 13 is about Heat Equation.
 Note 14 is about Wave Equation.
 Current Lecture Note > Note 15 is about Laplace Equation.
Quizzes
Exams
Supplementary Problems and Practice Exams
Homework
 Problem 1: Show that $e^{2x}+e^{2y}=1$ is an implicit solution to the DE $e^{xy}+e^{yx} \frac{dy}{dx}=0$.
 Problem 2: Find a 1parameter family of solutions of the DE $y'=y$ and the particular solution for which $y(3)=1$.
 Problem 3: Construct a direction field for the differential equation $y'=2x$.
 Problem 4: Find a particular solution to the DE $y'=e^{x+y}$ with the initial value $y(0)=0$.
 Problem 1: Check that if the differential equation is exact $(e^{x}\sin y+e^{y})dx  (xe^{y}  e^{x} \cos y) dy= 0$.
If it is exact then solve the differential equation.
 Problem 2: Check that if the differential equation is exact $e^{x}(x+1) dx +(ye^y  xe^x)dy= 0$.
If it is exact then solve the differential equation.
 Problem 3: Let $P(x)=\int p(x) dx$. Show that $e^{P(X)}$ is an integrating factor for the DE
\[
y'+p(x) y=q(x).
\]
 Problem 4: Suppose that $a,b,c,e$ are constants such that $aebc\neq 0$. Let $m$ and $n$ be arbitrary real numbers. Show that
\[
(ax^{m}y+by^{n+1})dx + (cx^{m+1}+exy^{n})dy=0
\]
has an integrating factor $\mu(x,y)=x^{\alpha}y^{\beta}$ for some $\alpha$ and $\beta$.
 Problem 1: Find the orthogonal trajectories of the family of circles centered on $x$axis and passing through the origin.
 Problem 2: Find the orthogonal trajectories of the family of curves having equation $e^x \cos(y)=k$.
 Problem 3: Find the general solution of the Bernoulli equation $xy'+y+x^2y^2e^x=0$.(You may need to rewrite the equation!).
 Problem 4: Find the general solution of the Bernoulli equation $x^2y'+2y=2e^{\frac{1}{x}}y^{\frac{1}{2}}$.
 Problem 5: Find the general solution of the Ricatti equation $y'=1+\frac{y}{x}\frac{y^2}{x^2}$ with given particular solution $y_1(x)=x$.
 Problem 6: Find the general solution of the Ricatti equation $y'=y^2+2xy+(x^21)$ with given particular solution $y_1(x)=x$.
 1. in Question 23.
 2., 3. 4. in Question 24.
 3. in Question 25.
 Question 26.
 Question 28.
 Find a power series solution $y(x)$ around the point
$x_0=0$ to the differential equation
\[
y''+y=0.
\]
Verify that the power series solution you found
has the form
\[
y(x)=a_0\cos(x)+a_1\sin(x).
\]
 By using the second method find at least first
four terms of the power series solution $y(x)$ around
the point $x_0=0$ to the differential equation
\[
y''=xy(y')^2
\]
with $y(0)=2$ and $y'(0)=1$ (assume that the solution is analytic around $x_0=0$).
 Consider the Rayleigh's equation
\[
my''+ky = ay'b(y')^{3}
\]
which models the oscillation of a clarinet reed.
Using the second method
find the first four terms of the power
series solution $y(x)$ around $x_0=0$ with
$m=k=a=1$ and $b=1/3$ with the initial conditions
$y(0)=0$ and $y'(0)=1$.
Write the first four terms of the solution $y(x)$.
 Consider the following differential equation
\[
y''+4 (y^{2}+1)y'+xy=0.
\]
Use the second method to find first
four terms of the power series solution
\[
y(x)=\sum\limits_{n=0}^{\infty} a_n x^n
\]
around $x_0=0$ with the initial conditions
$y(0)=0$ and $y'(0)=1$.
 For the following differential equation
\[
4xy''+2y'+y=0
\]
 Find and classify all points as ordinary, regular singular, or irregular singular points.
 For each of the regular point(s), find the corresponding indicial equation and find roots $r_1$ and $r_2$ of the indicial equation (Yes, there are two roots and the difference is not integer).
 Find the corresponding recurrence relations for each of the roots $r_1, r_2$.
 Find the corresponding power series solutions $y_1$ and $y_2$.
 For the following differential equation
\[
xy''+y'y=0
\]
 Find and classify all points as ordinary, regular singular, or irregular singular points.
 For each of the regular point(s), find the corresponding indicial equation and find the double root $r_1$of the indicial equation (Yes there is one double root).
 Find the corresponding recurrence relation for the root $r_1$.
 Find the corresponding power series solution $y_1$.
 Use the method of Frobenious and write down the general form of the second solution $y_2$.
 Find at least first two terms of the second solution $b_0$ and $b_1$.
 For the following differential equation
\[
xy''+y=0
\]
 Find and classify all points as ordinary, regular singular, or irregular singular points.
 For each of the regular point(s), find the corresponding indicial equation and find the roots $r_1$ and $r_2$ of the indicial equation (Yes there are two roots with $r_1r_2$ is integer).
 Find the corresponding recurrence relation for the roots $r_1$ and $r_2$.
 Find the corresponding power series solution for $y_1$.
 Use the method of Frobenious and write down the general form of the second solution $y_2$.
 Find at least first two terms of the second solution $b_0$ and $b_1$.
 Let $f(x)$ be given as
\[
f(x)=\left\{
\begin{array}{ll}
0 &\pi< x < 0, \\
x & 0< x< \pi.
\end{array}
\right.
\quad f(x)=f(x+2\pi).
\]
 Find the Fourier series $F(x)$ of $f(x)$.
 Using the first part verify that
\[
\frac{\pi}{4}=\sum\limits_{n=0}^{\infty} \frac{(1)^{n}}{(2n+1)}.
\]
 Using the first part verify also that
\[
\frac{\pi^2}{8}=\sum\limits_{n=0}^{\infty} \frac{1}{(2n+1)^{2}}.
\]
 Let $f(x)$ be given as
\[
f(x)=\left\{
\begin{array}{ll}
x &0\leq x\leq \frac{1}{2}\pi, \\
\pix & \frac{1}{2}\pi\leq x\leq \pi.
\end{array}
\right.
\]
 Extend $f(x)$ into an odd periodic function with period of $2\pi$ and find its Fourier series $F(x)$.
 Extend $f(x)$ into an even periodic function with period of $2\pi$ and find its Fourier series $F(x)$.
 Using either the first part or the second part verify that
\[
\frac{\pi^{2}}{8}=\sum\limits_{n=1}^{\infty} \frac{1}{(2n1)^{2}}.
\]

Problem 1: Consider the following Heat conduction problem
\[
\left\{
\begin{array}{l}
u_{xx}=u_{t}, \quad 0 < x < 2, \quad t > 0,\\
u(0,t)=0 \quad \mbox{and} \quad u(2,t)=0,\\
u(x,0)=3\sin(\pi x)4\sin(\frac{3\pi x}{2}).
\end{array}
\right.
\]
 By considering separation of variables $u(x,t)=X(x)T(t)$, rewrite the partial differential equation
in terms of two ordinary differential equations in $X$ and $T$ (take arbitrary constant as $\lambda$).
 Rewrite the boundary values in terms of $X$ and $T$.
 Now choose the boundary values which will not give a nontrivial solution and write the ordinary differential equation corresponding to $X$.
 Solve the twopoint boundary value problem corresponding to $X$. Find all eigenvalues $\lambda_n$ and eigenfunctions $X_n$.
 For each eigenvalue $\lambda_n$ you found, rewrite and solve the ordinary differential equation corresponding to $T_n$.
 Now write general solution for each $n$, $u_n(x,t)=X_n(x) T_n(t)$ and find the general solution $u(x,t)=\sum u_n(x,t)$.
 Using the given initial value and the general solution you found, find the particular solution.

Problem 2: Consider the following Heat conduction problem
\[
\left\{
\begin{array}{l}
9u_{xx}=u_{t}, \quad 0 < x < 3, \quad t > 0,\\
u_{x}(0,t)=0 \quad \mbox{and} \quad u_{x}(3,t)=0,\\
u(x,0)=2\cos(\frac{\pi x}{3}) 4 \cos(\frac{5\pi x}{3}).
\end{array}
\right.
\]
 By considering separation of variables $u(x,t)=X(x)T(t)$, rewrite the partial differential equation
in terms of two ordinary differential equations in $X$ and $T$ (take arbitrary constant as $\lambda$).
 Rewrite the boundary values in terms of $X$ and $T$.
 Now choose the boundary values which will not give a nontrivial solution and write the ordinary differential equation corresponding to $X$.
 Solve the twopoint boundary value problem corresponding to $X$. Find all eigenvalues $\lambda_n$ and eigenfunctions $X_n$.
 For each eigenvalue $\lambda_n$ you found, rewrite and solve the ordinary differential equation corresponding to $T_n$.
 Now write general solution for each $n$, $u_n(x,t)=X_n(x) T_n(t)$ and find the general solution $u(x,t)=\sum u_n(x,t)$.
 Using the given initial value and the general solution you found in, find the particular solution.

Problem: Consider the following Wave equation which describes the displacement $u(x, t)$ of a piece of flexible string with the initial boundary
value problem
\[
\left\{
\begin{array}{l}
25u_{xx}=u_{tt}, \quad 0 < x < 5, \quad t > 0,\\
u(0,t)=0 \quad \mbox{and} \quad u(5,t)=0,\\
u(x,0)=0 \quad \mbox{and} \quad u_{t}(x,0)=3\sin(\frac{3\pi x}{5})10\sin(\frac{4\pi x}{5}).
\end{array}
\right.
\]
 By considering separation of variables $u(x,t)=X(x)T(t)$, rewrite the partial differential equation
in terms of two ordinary differential equations in $X$ and $T$.
 Rewrite the boundary values in terms of $X$ and $T$.
 Now choose the boundary values which will not give a nontrivial solution and then rewrite the ordinary differential equation corresponding to $X$.
 Solve the twopoint boundary value problem corresponding to $X$ you found. Find all eigenvalues $\lambda_n$ and eigen functions $X_n$
 For each eaigenvalue $\lambda_n$ you found, solve the initial value problem corresponding to $T_n$.
 Now write general solution for each $n$, $u_n(x,t)=X_n(x) T_n(t)$ and find the general solution $u(x,t)=\sum u_n(x,t)$.
 Using the given initial values, find the particular solution.
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